We introduce a new class of possibly infinite dimensional Lie algebras and study their structural properties. Examples of this new class of Lie algebras are finite dimensional simple Lie algebras containing a nonzero split torus, affine and extended affine Lie algebras. Our results generalize well-known properties of these examples.
We construct certain integral structures for the cores of reduced tame extended affine Lie algebras of rank at least 2. One of the main tools to achieve this is a generalization of Chevalley automorphisms in the context of extended affine Lie algebras. As an application, groups of extended affine Lie type associated to the adjoint representation are defined over arbitrary fields.
In this paper, we study nullity-2 toroidal extended affine Lie algebras in the context of vertex algebras and their $ϕ$-coordinated modules. Among the main results, we introduce a variant of toroidal extended affine Lie algebras, associate vertex algebras to the variant Lie algebras, and establish a canonical connection between modules for toroidal extended affine Lie algebras and $ϕ$-coordinated modules for these vertex algebras. Furthermore, by employing some results of Billig, we obtain an explicit realization of irreducible modules for the variant Lie algebras.
In this note we explain, in terms of finite dimensional representations of Lie algebras $\mathfrak{sp}_{2\ell}\subset\mathfrak{sl}_{2\ell}$, a combinatorial coincidence of difference conditions in two constructions of combinatorial bases for standard representations of symplectic affine Lie algebras.
The interaction of a Lie algebra $\LL,$ having a weight space decomposition with respect to a nonzero toral subalgebra, with its corresponding root system forms a powerful tool in the study of the structure of $\LL.$ This, in particular, suggests a systematic study of the root system apart from its connection with the Lie algebra. Although there have been a lot of researches in this regard on Lie algebra level, such an approach has not been considered on Lie superalgebra level. In this work, we introduce and study extended affine root supersystems which are a generalization of both affine refl…
Feigin-Stoyanovsky's type subspaces for affine Lie algebras of type $C_\ell^{(1)}$ have monomial bases with a nice combinatorial description. We describe bases of whole standard modules in terms of semi-infinite monomials obtained as "a limit of translations" of bases for Feigin-Stoyanovsky's type subspaces.
In this paper, we study a family of infinite-dimensional Lie algebras $\widehat{X}_{S}$, where $X$ stands for the type: $A,B,C,D$, and $S$ is an abelian group, which generalize the $A,B,C,D$ series of trigonometric Lie algebras. Among the main results, we identify $\widehat{X}_{S}$ with what are called the covariant algebras of the affine Lie algebra $\widehat{\mathcal{L}_{S}}$ with respect to some automorphism groups, where $\mathcal{L}_{S}$ is an explicitly defined associative algebra viewed as a Lie algebra. We then show that restricted $\widehat{X}_{S}$-modules of level $\ell$ naturally co…
The aim of this work is to construct the quasi-particle basis of principal subspace of standard module of highest weight $kΛ_0$ of level $k\geq 1$ of affine Lie algebra of type $G_2^{(1)}$ by means of which we obtain the basis of principal subspace of generalized Verma module.
Relations for annihilating fields of standard modules for affine Lie algebras
J. Lepowsky和R. L.威尔逊通过仿射李代数的顶点算子构造，开创了组合罗杰斯-拉马努金型恒等式的方法。在一个联合工作与阿恩Meurman这种做法是进一步发展的框架内的顶点算子代数。该构造的主要成分是定义标准模块的关系以及它们之间的关系。讨论的内容涉及到表示论和组合数学，最后的结果只对仿射李代数$A_1^{(1)}$和$A_2^{(1)}$成立.在本文中的一些论点制定和推广一般仿射李代数。主要结果是一种秩定理，保证存在的组合关系之间的关系，提供了某些纯组合量等于某些表示空间的尺寸。虽然这一结果在相当一般的情况下成立，但其应用主要是仿射李代数的标准模。
Using the tensor category theory developed by Lepowsky, Zhang and the second author, we construct a braided tensor category structure with a twist on a semisimple category of modules for an affine Lie algebra at an admissible level. We conjecture that this braided tensor category is rigid and thus is a ribbon category. We also give conjectures on the modularity of this category and on the equivalence with a suitable quantum group tensor category. In the special case that the affine Lie algebra is $\widehat{\mathfrak{sl}}_2$, we prove the rigidity and modularity conjectures.
We construct a new family of irreducible modules over any basic classical affine Kac-Moody Lie superalgebra which are induced from modules over the Heisenberg subalgebra. We also obtain irreducible deformations of these modules for the quantum affine superalgebra of type A.
Using the bases of principal subspaces for twisted affine Lie algebras except $A_{2l}^{(2)}$ by Butorac and Sadowski, we construct bases of the highest weight modules of highest weight $kΛ_0$ and parafermionic spases for the same affine Lie algebras. As a result, we obtain their character formulas conjectured in arXiv:math/0102113.
In this paper, we explore natural connections among trigonometric Lie algebras, (general) affine Lie algebras, and vertex algebras. Among the main results, we obtain a realization of trigonometric Lie algebras as what were called the covariant algebras of the affine Lie algebra $\widehat{\mathcal{A}}$ of Lie algebra $\mathcal{A}=\frak{gl}_{\infty}\oplus\frak{gl}_{\infty}$ with respect to certain automorphism groups. We then prove that restricted modules of level $\ell$ for trigonometric Lie algebras naturally correspond to equivariant quasi modules for the affine vertex algebras $V_{\widehat{\…
Crystals for Demazure Modules of Classical Affine Lie Algebras
本文研究了$A^{(1)}_n,B^{(1)}_n,C^{(1)}_n,D^{(1)}_n, A^{(2)}_{2n-1},A^{(2)}_{2n}, and D^{(2)}_{n+1}$型仿射李代数的Demazure模在路径实现中的结晶。我们找到了一个特殊的序列的仿射Weyl群元素的选择完美的晶体，并显示，如果最高的权重是$l\La_0$，Demazure晶体具有一个非常简单的结构。
We construct combinatorial bases of principal subspaces of standard modules of level $k \geq 1$ with highest weight $kΛ_0$ for the twisted affine Lie algebras of type $A_{2l-1}^{(2)}$, $D_l^{(2)}$, $E_6^{(2)}$ and $D_4^{(3)}$. Using these bases we directly calculate characters of principal subspaces.
We construct quasi-particle bases of principal subspaces of standard modules $L(Λ)$, where $Λ=k_0Λ_0+k_jΛ_j$, and $Λ_j$ denotes the fundamental weight of affine Lie algebras of type $B_l^{(1)}$, $C_l^{(1)}$, $F_4^{(1)}$ or $G_2^{(1)}$ of level one. From the given bases we find characters of principal subspaces.
In this paper we give a combinatorial parametrization of leading terms of defining relations for level $k$ standard modules for affine Lie algebra of type $C_{n}\sp{(1)}$. Using this parametrization we conjecture colored Rogers-Ramanujan type combinatorial identities for $n\geq 2$ and $k\geq 2$; the identity in the case $n=k=1$ is equivalent to one of Capparelli's identities.
We construct a family of homomorphisms between Weyl modules for affine Lie algebras in characteristic p, which supports our conjecture on the strong linkage principle in this context. We also exhibit a large class of reducible Weyl modules beyond level one, for p not necessarily small.
In this paper we construct combinatorial bases of parafermionic spaces associated with the standard modules of the rectangular highest weights for the untwisted affine Lie algebras. Our construction is a modification of G. Georgiev's construction for the affine Lie algebra $\widehat{\mathfrak sl}(n+1,\mathbb C)$---the constructed parafermionic bases are projections of the quasi-particle bases of the principal subspaces, obtained previously in a series of papers by the first two authors. As a consequence we prove the character formula of A. Kuniba, T. Nakanishi and J. Suzuki for all non-simply-…
